A GENERALIZATION OF PRATT-ARROW MEASURE TO NON-EXPECTED-UTILITY PREFERENCES AND INSEPARABLE PROBABILITY AND UTILITY

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1 A GENERALIZATION OF PRATT-ARROW MEASURE TO NON-EXPECTED-UTILITY PREFERENCES AND INSEPARABLE PROBABILITY AND UTILITY Robert F. Nau Fuqua School of Busness Duke Unversty Durha, NC , USA February 26, 2003 Forthcong n Manageent Scence Abstract: The Pratt-Arrow easure of local rsk averson s generaled for the n-densonal state-preference odel of choce under uncertanty n whch the decson aker ay have nseparable probabltes and subjectve utltes, unobservable stochastc pror wealth, and/or sooth non-expected-utlty preferences. Local rsk averson s easured by the atrx of dervatves of the decson aker s rsk neutral probabltes, wthout reference to true subjectve probabltes or rskless wealth postons, and coparatve rsk averson s easured wthout requrng agreeent on true probabltes. Rsk neutral probabltes and ther dervatves are shown to be suffcent statstcs for approxately optal nvestent and fnancng decsons n coplete arkets for contngent clas. JEL Classfcaton: D80. Key Words: Crtera for Decson Makng under Rsk and Uncertanty; Rsk Averson; Uncertanty Averson; Expected-Utlty Theory; Non-Expected-Utlty; Sooth Preferences I a grateful to J Anton, Ed Karn, Pno Lopoo, Mark Machna, Aleksandar Pekec, J Sth, Peter Wakker; senar partcpants at the London School of Econocs, FUR IX, Pars-IX, Pars-I, and Duke Unversty; and several anonyous referees for coents on earler drafts. Ths research was supported by the Natonal Scence Foundaton under grant , by the Fuqua School of Busness, and by INSEAD.

2 . INTRODUCTION Rsk averson s coonly defned as a departure fro expected-value-axng behavor: a rsk averse person always prefers a rskless wealth poston to a rsky poston wth the sae expected value. For a decson aker wth expected-utlty preferences, deternate probabltes, state-ndependent utlty, and observable wealth, ths defnton s as good as any other, and the decson-aker s degree of rsk averson s convenently represented by the Pratt- Arrow easure de Fnett 952, Pratt 964, Arrow 965, whch quantfes the local curvature of her Bernoull utlty functon. Yet n any econocally nterestng stuatons nvolvng uncertanty, the defnton of rsk averson by reference to expected values and rskless wealth can be probleatc, especally fro the vewpont of an observer. Probabltes ay be subjectve and utltes ay be state-dependent, n whch case probabltes and expected values are not unquely revealed by preferences. If the ndvdual has non-expected utlty preferences e.g., f she exhbts averson to uncertanty her belefs ay not even be representable by addtve probabltes. Indvduals ay also have sgnfcant unobserved pror stakes n events and they ay face unnsurable rsks, n whch case rskless wealth postons ay be ll-defned or unattanable, and wthout ndependent knowledge of correlatons wth pror wealth t s not clear whether the acquston of another rsky asset produces an ncrease or decrease n overall rsk. Yaar 969 has suggested a ore eleentary defnton of rsk averson that does not depend on observablty of probabltes or pror wealth, naely that a decson aker s rsk averse f her preferences are payoff-convex, that s, convex wth respect to deternstc xtures of payoffs wthn states of the world. The latter defnton, whch we shall adopt here, agrees wth the conventonal one for decson akers who are expected-utlty axers, and t also apples n a straghtforward way to decson akers wth state-dependent

3 utlty or non-expected-utlty preferences, although t s not the only possble defnton of rsk averson n the latter cases e.g., Machna 995, Epsten 999. Ths paper derves a atrxvalued easure of rsk and uncertanty averson that generales the Pratt-Arrow easure to the broader fraework of Yaar s defnton. To otvate the dscusson, Fgure shows the ndfference curves of four hypothetcal ndvduals wth respect to dstrbutons of wealth over two states of the world. The curves are algned so that the orgn of coordnates corresponds to the status quo for each ndvdual, although t s not a rskless poston for anyone: they all have stochastc pror wealth. Alce clas to be an expected utlty axer havng the utlty functon Uw = p exprw p exprw 2, wth p = /3, r = 0.2, and pror wealth dstrbuton w, w 2 = 2.8, 4.3. Thus, Alce exhbts constant absolute averson to rsk wth a rsk averson coeffcent of 0.2, and she evdently assgns probablty /3 to state. Bob has a Cobb-Douglas utlty functon, Uw = α w w α 2, wth α = and pror wealth dstrbuton 2.5, 3.0. Of course, ths s equvalent to havng expected-utlty preferences wth logarthc utlty and a probablty of for state, but Bob nssts that he does not subscrbe to expected-utlty theory and has no opnon concernng the probabltes of the states; hs preferences just are what they are. Carol has state-dependent expected utlty preferences represented by the utlty functon Uw = p expr w pβ expr 2 w 2 wth r =, r 2 = Thus, Carol s uch ore rsk averse than Alce n state as easured by concavty of utlty for state- wealth, whle she s less rsk averse n state 2. However, she s evasve about her other utlty paraeters: she says t s Rsk averson concepts for non-expected-utlty preferences under rsk are dscussed by Chew et al. 987, Chew 995, Cohen 995, and Courtault and Gayant 998. Here the focus s on choce under uncertanty, where probabltes of events are at best subjectve and at worst ndeternate. 2

4 possble that her probablty of state s p =/3, her rate of utlty substtuton between states s β =, and her pror wealth s 2.8, 2.0. However, t could be that p = /2 and β = 2. Or, perhaps her pror wealth s 3.023, 2.0 whle p =/3 and β = 0.8. As she s fond of pontng out, t really doesn t atter: her preferences for changes n wealth would be the sae n all three cases. Carol also suspects that Alce has been less than truthful: she thnks that Alce has addtonal, undsclosed wealth n state and that Alce s true probablty for state s really greater than /3, but she can t prove t erely by watchng how Alce bets or trades wth others. Fnally, Dan has non-expected utlty preferences represented by the utlty functon Uw = exprpw +pw 2 exprqw +qw 2, wth p = 0.05, q = 0.95, r = 2, and a pror wealth dstrbuton of.25,.0. 2 Concdentally, everyone has the sae argnal rate of substtuton for wealth between states and 2, naely that $3 n state s equvalent to $2 n state 2 n other words, the rsk neutral probablty of state s 0.4 for everyone perhaps because they have already engaged n trade wth each other, or perhaps because they are n contact wth an external arket n whch the prce of a state- Arrow securty s 0.4. Suppose that we wsh to easure and copare the degrees of local rsk averson of these four ndvduals n order to deterne ther relatve propensty to gable, purchase rsky assets, or take other decsons under uncertanty, startng fro ther status quo wealth postons rather than hypothetcal rskless postons. The usual defntons of rsk averson and rsk prea do not apply, snce probabltes and current wealth postons are ll-defned or unobservable, and only 2 A utlty functon of ths for can be used to odel uncertanty averson and ratonale the Ellsberg and Allas paradoxes Nau 2002, Klbanoff et al Here, t s as f Dan has lnear utlty for oney but he s uncertan about the probablty of state, whch he feels s equally lkely to be ether 0.05 or 0.95, and he has constant absolute averson to uncertanty wth an uncertanty averson coeffcent of 2. 3

5 Alce s local rsk preferences can be charactered by a scalar Pratt-Arrow easure. Yet all four are planly rsk averse n the sense of havng preferences that are payoff-convex, and they can even be strctly ordered n ters of ther degrees of local rsk averson: Alce<Bob<Carol<Dan. The ndfference curves passng through ther status quo wealth postons are the fronters of ther respectve sets of acceptable gables whch are observable and followng Yaar 969, an ndvdual wth a strctly saller set of acceptable gables s ore rsk averse. To copare local rsk preferences n these ters, t s unnecessary to know anyone s probabltes, utlty functons for oney, or pror wealth: t suffces to consder the slope and curvature of ther respectve ndfference curves. Yaar explored the two-densonal case, showng that the degree of local rsk averson could be easured by the second dervatve of the paraetered ndfference curve. However, the latter easure does not generale easly to hgher densons. Ths paper derves a easure of local rsk averson for the general n-densonal case, n whch the curvature of ndfference curves s easured by the atrx of dervatves of the decson aker s rsk neutral probabltes.e., the dervatves of the noraled gradent of her ordnal utlty functon. In the specal case where preferences are separable across utually exclusve events, the new rsk averson easure s addtvely separable and reduces to a statedependent for of the Pratt-Arrow easure, n whch case the decson aker s uncertanty neutral. 3 For such an ndvdual, the rsk neutral dstrbuton s the correct probablty 3 A decson aker s uncertanty averse f she dslkes bettng on events wth abguous probabltes, as n Ellsberg s paradox, and uncertanty neutral otherwse. Second-order averson to rsk or uncertanty arses fro the curvature of ndfference curves, whereas frst-order averson arses fro knks n ndfference curves Segal and Spvak 990. In the Choquet and axn expected utlty odels, uncertanty averson s exclusvely a frstorder effect, whereas the focus of ths paper s on second-order effects.e., sooth rather than knked preferences. 4

6 dstrbuton to use n conjuncton wth the Pratt-Arrow easure when calculatng rsk prea n the presence of background rsk and/or state-dependent utlty. Under ore general preferences, the new easure ncorporates both second-order rsk averson and second-order uncertanty averson. A non-neutral atttude toward uncertanty s revealed when a decson aker s unforly ore rsk averse toward bets on soe events than toward others, as dscussed n a copanon paper Nau The organaton of the paper s as follows. Secton 2 presents the odelng fraework and defntons of rsk averson and rsk prea. Secton 3 derves the an result, naely a generalaton of the Pratt-Arrow easure for the general n-state odel. Secton 4 consders the specal case of separable preferences essentally, state-dependent subjectve expected utlty wthout unquely deterned probabltes for whch the rsk averson easure s vectorvalued. Secton 5 dscusses coparatve rsk averson and argnal nvestent behavor; secton 6 consders decson-akng n arkets under uncertanty; secton 7 apples the arket results to a project valuaton decson; and secton 8 presents concludng coents. 2. THE MODEL The analytc fraework used here s that of state-preference theory Debreu 959, Arrow 964, Hrshlefer 965, whch ncludes, as specal cases, odels of expected utlty, subjectve expected utlty, state-dependent utlty, and Choquet and axn expected utlty as they apply to onetary acts. In the state-preference fraework, objects of choce are dstrbutons of onetary wealth over states of the world, represented by vectors n R n. The decson aker s preferences aong such wealth dstrbutons are nally assued to satsfy the basc axos of consuer theory copleteness, transtvty, contnuty, whch ply that they can be represented by an ordnal utlty functon U and vsualed n ters of ndfference curves n 5

7 payoff space, as llustrated n Fgure. In general, U s deterned only up to onotonc transforatons and s therefore unobservable. A decson aker who s rsk neutral has lnear ndfference curves n payoff space, whle one who s rsk averse does not. The conventonal defnton of rsk averson, naely that a rsk averse ndvdual always prefers a rskless wealth poston to a rsky poston wth the sae expected value, has the followng geoetrc nterpretaton: an ndfference curve drawn through a pont on the 45-degree certanty lne n payoff space ust le on or above the so-expectedvalue lne through the sae pont. A stronger defnton, drawng on Rothschld and Stglt s 970 concept of ncreasng rsk, s that a rsk averse ndvdual dslkes ean-preservng spreads n payoff dstrbutons, whch eans that any oveent along an so-expected value lne away fro the 45-degree certanty lne.e., n a drecton that takes payoffs n all states farther fro the expected value s dspreferred to the status quo, even f the status quo s already rsky. A thrd defnton of rsk averson averson, due to Yaar 969 s that a rsk averse ndvdual has preferences that are payoff-convex 4, whch ples that her ordnal utlty functon U s quas-concave. The three defntons are equvalent for decson akers wth expected-utlty preferences, 5 but they dffer for decson akers wth non-expected utlty preferences. Although the ean-preservng-spread defnton characteres a for of local rsk averson, n the sense that the defnton has local plcatons even at rsky wealth postons, t nevertheless adts local behavor that s arguably rsk-lovng. In partcular, t adts the possblty that for soe 4 The preference relaton s [strctly] payoff-convex f x and y ply αx+αy [>] for α 0,. 5 For a decson aker wth expected utlty preferences, local rsk averson at rskless wealth postons s suffcent to ensure concavty of the Bernoull utlty functon the unvarate utlty functon for oney, whch ples that preferences are everywhere locally rsk averse and also payoff-convex. 6

8 rsky wealth poston w and fnte gable, both w+ and w are strctly preferred to w. For an llustraton, see Machna 995; see also Karn 995. Thus, for exaple, the decson aker ght be wllng to pay a fee for the prvlege of placng a bet on an event and also alternatvely wllng to pay a fee for the prvlege of placng the very opposte bet. Ths sort of behavor, whch resebles gablng-for-ts-own-sake, s expressly forbdden by Yaar s defnton. On the other hand, the ean-preservng spread defnton requres pror wealth to be observable so that the certanty lne can be located, and t requres the decson aker to be probablstcally sophstcated Machna and Schedler 992 so that expected values are unquely deterned, whereas Yaar s ethod akes no such requreents. For these reasons, Yaar s defnton of rsk averson as payoff-convexty of preferences or equvalently, as quas-concavty of utlty wll be adopted henceforth. Assue that preferences over wealth dstrbutons are sooth, so that the utlty functon U that represents the s twce dfferentable. 6 By vrtue of onotoncty, the gradent of U at w s a non-negatve vector and t can be noraled to yeld a probablty dstrbuton: π w U w U w = U w. U w π w s nvarant to onotonc transforatons of U and s observable. It s coonly known as a rsk neutral probablty dstrbuton because the decson aker prces very sall assets n a seengly rsk-neutral anner wth respect to t. More precsely, let denote the payoff vector of a rsky asset and let P; w denote the argnal prce that the decson aker s wllng to pay 6 The soothness assupton rules out Choquet or axn expected utlty preferences, but t perts non-expected utlty preferences that are arbtrarly close to Choquet expected utlty n the fashon of Dan s preferences n Fg.. 7

9 for at wealth w, n the sense that she s wllng to pay αp; w to receve α n the lt as α goes to ero. Then P; w s deterned by: l α 0 Uw+α αp; w Uw/α = 0, for whch the frst-order condton s P; w = π w Eπw[]. Hence, the argnal prce s the rsk neutral expectaton of the asset under the local rsk neutral dstrbuton π w. wll be sad to be a neutral asset at the current wealth poston f Eπw[] = 0. The argnal prce of s the per-unt prce at whch the decson aker would buy or sell an nfntesal share. Her buyng prce for n ts entrety, denoted B; w, s deterned by Uw+ B; w Uw = 0, whle her sellng prce C; w, otherwse known as her certanty equvalent for, satsfes Uw+ Uw+C; w = 0. The buyng and sellng prces are generally slar, but not dentcal, as llustrated n Fgure 2, and they are related by C; w = B; w+. The functonal dependence of π w on w reveals the decson aker s atttude toward rsk and uncertanty. If the decson aker s rsk neutral.e., f U s both quas-concave and quas-convex, then she has lnear ndfference curves, π w s constant and P; w = B; w = C; w at all w. If she s rsk averse, π w vares wth w accordng to the local curvature of the ndfference curves. Intutvely, a decson aker who s rsk averse has dnshng argnal utlty for all rsky assets, hence her buyng and sellng prces for an asset wll typcally be less than ts argnal prce. To ake ths noton precse, let the buyng rsk preu assocated wth at wealth w, here denoted b; w, be defned as the dfference between the asset s argnal prce and ts buyng prce: 8

10 b; w = E πw [] B; w. The sellng rsk preu 7 s slarly defned by c; w = E πw [] C; w. Refer agan to Fgure 2. Pratt s 964 rsk preu s the specal case of the sellng rsk = = n j j w j preu that obtans when U has the expected-utlty representaton U w p u, where u s a twce-dfferentable state-ndependent utlty functon for oney, p s a known probablty dstrbuton, and the decson aker begns n a state of rskless wealth w = x, whch s a pont on the 45-degree certanty lne n payoff space. If the decson aker s a possbly state-dependent expected-utlty axer, her rsk neutral probabltes are sply the product of her true probabltes and relatve argnal utltes for oney. That s, π j w p j u j w j, where u j s the frst dervatve of the utlty functon for oney n state j Drèe 970. Under Pratt s assuptons of state-ndependent utlty and rskless pror wealth, true probabltes and rsk neutral probabltes happen to concde, and the sellng rsk preu can be nterpreted as the aount of expected value the decson aker would gve up to elnate all rsk followng the nvoluntary acquston of. The falar result s that f s sall and actuarally neutral E p [] = 0, then the sellng rsk preu s approxately c; x ½ rxe p [ 2 ] = ½ rxvar p [] = ½ Cov p [, rx], 7 Buyng and sellng rsk prea are called copensatng and equvalent rsk prea by Kball 990, who ponts out that they are essentally equvalent for sall rsks under state-ndependent expected utlty and rskless pror wealth. In the context of state-dependent utlty, Karn 983, 985 uses a generalaton of the sellng preu, whle Kelsey and Nordqust 99 prefer the buyng preu, notng that the sellng preu creates techncal probles under soe state-dependent utlty functons. 9

11 where p = π x s the true dstrbuton that concdes wth the rsk neutral dstrbuton and rx = u x/u x s the Pratt-Arrow easure of absolute rsk averson. The buyng rsk preu converges to the sae lt when s sall enough for to be accurate, as wll be seen. Thus, ½ rx s the decson aker s local prce of rsk, where rsk s easured n ters of the varance of the asset, and the sellng rsk preu s non-negatve at constant wealth level x f and only f rx s non-negatve. The functon rx also perts coparatve rsk averson to be charactered n a sple way: f rx and r 2 x are the rsk averson easures of agents and 2, respectvely, then agent s as rsk averse as agent 2 n the sense of assgnng greater or equal rsk prea f r x r 2 x for all x, and Pratt and Arrow showed that n ths case agent wll also nvest less than agent 2 n a rsky asset when gven a choce between a sngle rsky asset and a safe asset. The buyng rsk preu rather than the sellng rsk preu wll be used henceforth as the yardstck for easurng local rsk averson, for several reasons. Frst, the queston that the sellng preu s desgned to address, naely how uch expected value the decson aker would gve up to elnate all the rsk she currently faces, s oot n the present settng of unobservable stochastc pror wealth and state-dependent or non-expected utlty preferences. It s ore natural to ask how uch addtonal argnal value the decson aker would requre as copensaton for takng on a new rsk. Second, the buyng rsk preu has the convenent property that b+x; w = b; w for any constant x, whch s not true for the sellng rsk preu except n specal cases. Thrd and ost portantly, the buyng rsk preu s a ore natural ndcator of rsk averson n the general state-preference fraework because t drectly easures the local quas-concavty of utlty. In partcular, Proposton : The decson aker s rsk averse f and only f her buyng rsk preu s non-negatve for every asset at every wealth dstrbuton. 0

12 Proof: Non-negatvty of the buyng rsk preu s essentally a defnton of quasconcavty: a functon U s quas-concave f and only f U w w w 0 whenever Uw Uw e.g., theore M.C.3 n Mas-Colell et al. 995, and f U s onotonc, t suffces for ths to hold when Uw = Uw. Lettng w = w + B; w, we have U w B; w 0, whch s equvalent to E πw [] B; w, whch n turn s equvalent to b; w 0. Q.E.D 3. A GENERAL MEASURE OF RISK AVERSION The objectve of ths secton s to charactere rsk averson n ters of second-order propertes of preferences. As s well known, U s quas-concave, and hence the decson aker s rsk averse by our defnton, f and only f at every w the Hessan atrx D 2 Uw s negatve sedefnte n the subspace of neutral assets. Ths s not edately helpful or econocally sgnfcant, however, because D 2 Uw s not observable that s, t s not unquely deterned by preferences. The observable second-order nforaton resdes nstead n the atrx of dervatves of the rsk neutral probabltes, D πw, whose jk th eleent s Dπ jk w = πjw/ w k = n 2 2 D U jk w π j w D Uhk w h= U w. 2 where D 2 U jk w denotes 2 Uw/ w j w k. In prncple, the eleents of D πw could be drectly easured by askng the decson aker to conteplate sall changes n her wealth n each state and to assess how her rsk neutral probabltes.e., her bettng rates on ndvdual states would change as a result. If w k s ncreased by a sall aount w k, the decson aker s rsk neutral probablty n state j ncreases by Dπ jk w wk + o w k, ceters parbus; and when total wealth changes fro w to w+ w, her rsk-neutral probablty dstrbuton changes fro π to π + π,

13 where π = D πw w + o w. D πw s generally asyetrc and has less than full rank. In partcular, ts coluns to ero, whch guarantees that the soluton to π = D πw w satsfes n j = π j = 0, a necessary condton for no-arbtrage. The an result of ths secton s that the decson aker s local and global atttudes toward rsk and uncertanty are copletely suared by D πw, generalng the statendependent expected utlty analyss of Pratt-Arrow and the 2-densonal state-preference analyss of Yaar: Proposton 2: a The rsk preu of a sall neutral asset satsfes b; w ½ D πw ; b The rsk preu of a sall non-neutral asset satsfes b; w ½ Q; w, where Q ; w π w D π w πw ; c The rsk preu of any asset satsfes b ; w = x Q ; w + x B x; w dx ; and consequently 0 d The decson aker s rsk averse f and only f, at every w, D πw s negatve sedefnte n the subspace of neutral assets. A detaled proof s gven n the appendx, but an nforal drect proof of parts a and b wll be sketched here. If s neutral E πw [] = 0, ts buyng prce s b; w by defnton. If the decson aker buys t at ths prce, thus keepng her utlty constant, her fnal wealth wll dffer fro her ntal wealth by the vector aount +b; w and the frst-order change n her rsk neutral probablty dstrbuton wll be D πw +b; w. Suppose that she purchases n 2

14 sall, equal ncreents at the prevalng argnal prces, thus holdng her utlty roughly constant. In the process, her rsk neutral dstrbuton wll change lnearly fro π to π + D πw +b; w and her argnal prce of wll ncrease lnearly fro 0 to D πw +b; w as her wealth follows a locally quadratc trajectory along an ndfference curve. The average argnal prce s the dpont, naely ½ D πw +b; w, whch s also the total prce by the lnearty of the prce trajectory, hence the rsk preu satsfes: b; w ½ D πw +b; w. If s suffcently sall n partcular, f D πw << then the factor of b; w nsde the parentheses on the RHS s nsgnfcant, whence b; w ½ D πw. If s not neutral, the ore general for b; w ½ Q; w of part b follows fro the dentty b+x; w = b; w wth x = π w. Thus, n the n-densonal state-preference fraework, the expresson rxe p [ 2 ] n Pratt s rsk preu forula for a neutral asset s replaced by the ore general atrx expresson D πw. Evdently D πw encodes both the decson aker s belefs and local rsk preferences, and ndeed t can be factored nto a product of two atrces, one of whch contans the decson aker s rsk neutral probabltes and the other of whch s constructed fro ratos of second and frst dervatves of the ordnal utlty functon, generalng the Pratt- Arrow easure. To show ths, defne the local rsk averson atrx as the atrx Rw whose jk th eleent s the followng rato of second to frst dervatves: r jk w = 2 Uw/ w j w k / Uw/ w j. 3

15 Under expected-utlty preferences, Rw would be an observable dagonal atrx, and wth state-ndependent utlty and rskless pror wealth w = x, the dagonal eleents would be r jj x = rx for every j, as wll be dscussed n ore detal n the followng secton. But under general preferences, Rw s nether a dagonal atrx nor s t observable, snce t s not nvarant to onotonc transforatons of U. In partcular, f U ˆ w = fuw, where f s onotonc and twce dfferentable, then U ˆ represents the sae preferences as U, but the correspondng rsk averson atrx Rˆ w dffers fro Rw by an addtve constant n each colun: Rˆ w = Rw + α Π w, T where Π w s the atrx whose rows are all equal to π w,.e., the atrx whose eleents n the k th colun are all equal to π k w; and α = Uwf Uw/f Uw where f and f are the frst and second dervatves of f. To elnate the arbtrary constants, let a noraled rsk averson atrx R w be defned by R w = Rw Π wrw. The jk th eleent of R w s then r jk w = r jk w E πw [r k w], 3 where r k w denotes the k th colun of Rw. It follows that E πw [ R w w] = 0 for any vector w. The noraled rsk averson atrx s nvarant to onotonc transforatons of U and s observable. Coparson of ters n 2 and 3 reveals that D πw and R w are related by D πw = Πw R w, 4 4

16 where Πw = dag π w. Fro 4 t s seen that the jk th eleent of R w s π j w/ w k /π j w, whch s nus the relatve rate of change of the rsk neutral probablty of state j as wealth ncreases n state k. In these ters, we have Corollary 2.: The rsk preu of a sall neutral asset satsfes b; w ½ Πw R w = ½ Πw R w = ½ Cov πw [, R w] Proof: The frst dentty follows fro 4. The second follows fro the fact that R can be substtuted for R when s neutral, as t dffers only by the colunwse addton of constants that drop out when t s preultpled by Πw. Q.E.D. Coparson wth shows that, under general condtons, the true dstrbuton p n the rsk preu forula s replaced by the local rsk neutral dstrbuton π w, whle the scalar Pratt-Arrow easure rx s replaced by the atrx rsk averson easure Rw, or equvalently by ts noraled, observable for R w. 4. THE SPECIAL CASE OF SEPARABLE PREFERENCES STATE-DEPENDENT UTILITY In the specal case where the decson aker has preferences that are separable across utually exclusve events.e., preferences that satsfy the ndependence axo 8 her ordnal utlty functon has the addtvely separable representaton: 8 Let x A, y ~A denote the wealth dstrbuton that agrees wth x n event A a subset of states and agrees wth y otherwse. The ndependence axo, whch s Savage s postulate P2, requres that x A, y ~A x A, y ~A f and only f x A, y ~A x A, y ~A for all x, y, x, y, and every event A. In other words, f two wealth dstrbutons agree on soe subset of states, then the drecton of preference between the doesn t depend on how they agree there. A behavoral volaton of the ndependence axo could be due to a dslke of abguous probabltes as n Ellsberg s paradox or soe other cause e.g., an attracton to sure thngs, as n Allas paradox. 5

17 Uw = v w + + v n w n. Debreu 960, Fshburn and Wakker 995 Ths representaton of preferences s equvalent to subjectve expected utlty wth state-dependent utltes and not-necessarly-unque subjectve probabltes, for t s always possble to wrte v j w j = p j u j w j where the nubers {p j } are arbtrarly-chosen probabltes sung to and the functons {u j w j } are correspondngly scaled state-dependent utltes. 9 Because events are not unquely ordered by probablty under ths representaton, separablty of preferences s not suffcent for probablstc sophstcaton, whch Epsten 999, also Epsten and Zhang 200 has equated wth uncertanty neutralty n a Savage-act fraework. But n the present fraework, separablty s suffcent for uncertanty neutralty because t ensures that preferences have an expected-utlty representaton even f t s not unque. When U s addtvely separable, ts cross-dervatves are ero and Rw = dagrw, where rw s a vector-valued Pratt-Arrow easure of rsk averson whose j th eleent s r j w = 2 Uw/ w j 2 / Uw/ w j = u j w j /u j w j, 5 and u j s the Bernoull utlty functon for oney n state j n an arbtrary expected-utlty representaton. The confounded probabltes and utlty-scale factors convenently drop out when r s coputed. Correspondngly, the eleents of R w satsfy r jk w = r k w jk π k w, whch can be nverted to obtan r j w = r jj w r kj w for j k, whence rw, lke R w, s 9 Even under Savage s axos, preferences aong ateral acts do not unquely deterne subjectve probabltes when utltes are potentally state-dependent, a troublesoe ssue that has been dscussed by Auann 97, Shafer 986, Karn and Mongn 2000, and Nau 200 aong others. Notwthstandng, Karn 983, 985 and Kelsey and Nordqust 99 requre that probabltes be unquely deterned for purposes of defnng rsk averson under state-dependent utlty, e.g., by the ethod of Karn et al

18 drectly observable. The atrx D πw of dervatves of the rsk neutral probabltes has generc eleent D π jk w = πjwr k w jk π k w and the rsk preu approxaton forula s accordngly specaled as: Corollary 2.2: For a decson aker wth separable preferences, the rsk preu of a sall neutral asset s: b; w ½ E πw [r w 2 ] = ½ Cov πw [, r w]. It follows that a suffcent condton for the decson aker to be rsk averse s r w 0 at every w, and a necessary condton s that at ost one eleent of rw ay be negatve. 0 By takng expectatons wth respect to the observable rsk neutral dstrbuton rather than the unobservable true dstrbuton, the probles of stochastc pror wealth and state-dependent utlty have been fnessed away: the decson aker s true probabltes and the correlatons between the rsky asset and her pror wealth are rrelevant once π w and r w have been observed. In the specal case where the Pratt-Arrow easure s constant across states e.g., f the decson aker has state-ndependent exponental utlty, Rw dagr where r s a scalar, n whch case D πw s syetrc and has ero row sus as well as ero colun sus. The rsk averson easure then can be taken outsde the expectaton n the approxaton forula for the local rsk preu: 0 It s perssble for the Pratt-Arrow easure to be negatve n one state because a gable that yelds a non-ero payoff n that state ust also yeld a non-ero payoff n one or ore other states, and f the Pratt-Arrow easure n all other states s suffcently postve, the net effect s stll rsk averson. 7

19 b; w ½ r E πw [ 2 ] = ½ r Var πw []. But even here, the local rsk neutral dstrbuton π w, rather than the true dstrbuton p, s used to evaluate the varance when argnal utltes vary across states due to pror stakes. The exact rsk preu slarly satsfes π x B x ; ] 0 w + w b ; w = r x Var [ dx, whch s a weghted average of ½ r Var π. [] along the ndfference curve through w, so that the rsk preu s transparently the prce of varance ½ r ultpled by a weghted average of the rsk neutral varance of n the vcnty of current wealth. The results of ths secton can be suared as follows: f the decson aker has convex preferences that are separable across states, her local preferences under uncertanty are copletely charactered by at ost a par of nubers for every state: a rsk neutral probablty and a rsk averson coeffcent. Such a person s rsk averse but uncertanty neutral. The rsk averson coeffcents ay be state-dependent, but unlke subjectve probabltes or utltes they are unquely deterned by preferences and hence they are observable. If separablty does not hold, the ore general atrx representaton of Proposton 2 apples, and lke Dan n Fgure the decson aker ay exhbt averson to uncertanty as well as rsk. 5. MARGINAL INVESTMENT AND COMPARATIVE RISK AVERSION Under state-ndependent expected-utlty preferences, t s possble to descrbe coparatve rsk averson and coparatve nvestent behavor n ters of propertes of the Bernoull utlty functon, as suared by ts Pratt-Arrow easure. Naturally, agent s ore rsk averse than agent 2 f r x > r 2 x at every level of rskless wealth x, where r s the Pratt-Arrow easure of agent. If agent s ore rsk averse than agent 2 n ths sense, she wll purchase saller quanttes of any rsky asset than agent 2 when both start fro the sae rskless ntal 8

20 wealth poston. When pror wealth s rsky stochastc, the stuaton s ore coplcated, and stronger notons of coparatve rsk averson and restrctons on the jont dstrbutons of old and new rsks are needed to obtan slar results. The ost tractable and thoroughly-studed case s that of probablstc ndependence between pror wealth and new rsks e.g., Khlstro et al. 98, Pratt 988, Goller and Pratt 996, Eeckhoudt et al. 996; a dfferent approach s taken by Ross 98. Sgnfcantly, when two expected-utlty-axng decson akers wth the sae probablstc belefs conteplate the sae rsky asset that s ndependent of ther pror wealth, they wll agree on the rsk neutral dstrbuton of the new asset: ther rsk neutral dstrbutons for the asset wll sply concde wth ts assued true dstrbuton, because ther expected argnal utltes for oney wll not depend on the new asset s value. In the uch ore general settng consdered n ths paper, t s not possble to surgcally reove a Bernoull utlty functon fro the decson aker: rsk prea and nvestent behavor ay depend on unobserved stochastcty of pror wealth, state-dependence of utlty, and averson to uncertanty as well as rsk. Nevertheless, n the sprt of Yaar s characteraton of ore rsk averse, the local and non-local rsk preferences of dfferent decson akers can be copared n ters of the curvature of ther ndfference curves for wealth, as quantfed by the atrces of dervatves of ther rsk neutral probabltes n approprate neghborhoods of the status quo. It s not necessary for decson akers to agree on the probablty dstrbutons of assets or even to have probablstc belefs although when coparng rsk atttudes toward a partcular asset t wll be necessary to assue that they ntally agree on the asset s argnal prce.e., ts rsk neutral expectaton. They can always reach such an agreeent, f necessary, by tradng the asset between the, even f other rsks that they face are not nsurable. 9

21 The local curvature of ndfference curves n the drecton of s easured by the quadratc for D πw. Optal purchases of a sngle rsky asset are nversely proportonal to ths easure of curvature, approprately averaged, as shown n the followng: Proposton 3: If a rsk averse ndvdual has the opportunty only to purchase shares of a new asset wth net payoff vector havng a postve argnal prce P; w = π w > 0 : a she wll optally purchase a quantty α that satsfes α Dπ w + x dx = 0 π w, and b f π w π w s suffcently sall, the optal quantty satsfes α. Dπ w Proof: For part a, note that the argnal prce of ust be ero followng an optal purchase. As wealth changes fro w + x to w + x+dx, the rsk neutral dstrbuton changes by D π w + x dx, and the argnal prce of changes by D π w + x dx. Hence the total change n argnal prce when α s acqured s α Dπ w + x 0 dx, and at the optal value of α, ths quantty ust equal πw. The approxaton forula n part b apples when the relatve argnal prce π w /, whch s the sne of the angle between and ts projecton n the subspace of neutral assets, s sall enough for D πw to be effectvely constant over the range of ntegraton. Corollary 3.: Suppose that two ndvduals wth wealth, prces, rsk prea, etc., subscrpted by {, 2}, have the opportunty to purchases shares of a new asset wth net payoff vector such that P ; w = P 2 ; w 2 > 0, and ndvdual optally purchases quantty α. 20

22 Then ndvdual 2 wll optally purchase no less than ndvdual f D π w + x Dπ w + for all x α x Corollary 3.2: If π w = π and D π w D π s negatve sedefnte, 2 w 2 2 w2 then ndvdual 2 wll purchase no less than ndvdual of any asset for whch π w / s suffcently sall. If both agents have separable preferences, a suffcent condton for the negatve sedefnteness requreent s r w r 2 w 2 pontwse, where r s the vector rsk averson easure of agent. 6. MARKETS UNDER UNCERTAINTY If the decson aker s ebedded n a arket for contngent clas where the relatve prces of assets are suared by a arket rsk neutral dstrbuton π *, the frst-order condton of utlty axaton s that her own rsk neutral probabltes should equlbrate wth those of the arket,.e., π w = π *. 2 Under these condtons, her responses to sall changes n wealth or prces are copletely deterned by π w and D πw. For exaple, suppose the decson aker receves a lup-su aount of ncoe x. It s of nterest to deterne the change w n her state-contngent wealth that wll obtan after she has re-equlbrated wth the arket, holdng prces fxed. Ths vector les along the wealth expanson path eanatng fro the decson These quadratc fors are typcally negatve for rsk averse ndvduals, so the nequalty n Corollary 3. eans that the LHS s ore negatve and hence larger n agntude than the RHS. The sense of both corollares s that ndvdual s ore rsk averse than ndvdual 2 f D s ore negatve sedefnte than D π n an approprate neghborhood of current wealth. π 2 2 Schlee and Schlesnger s 993 concept of a generaled rsk preu s defned n ters of the value to the decson aker of the opportunty to trade contngent clas at exogenous arket prces. 2

23 aker s current wealth dstrbuton, and t s the soluton of the equatons D π w w = 0 rsk neutral probabltes rean unchanged and π w w = x the arket value of the wealth change s x. Another quantty of nterest s the self-fnanced change n wealth w that wll be observed f the arket s rsk neutral probabltes change by a sall aount π, analogous to the Slutsky equaton of consuer theory. The change n wealth s the soluton to D π w + w w = π rsk neutral probabltes change by 2 D π between w and w+ w and π w + change s ero. These detals of these solutons are gven by Proposton 4: In a coplete arket for contngent clas: π, usng the dpont value of π w = 0 the new arket value of the wealth a the change n wealth nduced by a sall lup-su ncoe x, holdng prces fxed, s w R w x, Eπ w [ R w] hence R w s the drecton of the wealth expanson path at w; and b the self-fnanced change n wealth nduced by a sall change π n arket rsk neutral probabltes s w R π w π w π y, where π Eπ w + π R w π w + 2 π y. Eπ w + π [ R w] Proof: For part a, recall that D πw = ΠwRw Π wrw and verfy by substtuton that the wealth-expanson condton, D πw w = 0, s satsfed by w R w. The arket value condton, π w w = x, s then satsfed by choosng x/e πw [R w] as the scale factor. For part b, note that the ter nvolvng y s n the drecton of the wealth expanson path by part a, so t has no effect on rsk neutral probabltes, and y s deterned 22

24 precsely so that the self-fnancng condton, π w + π w = 0, s satsfed. It reans to show that the ter nvolvng π / π w+½ π yelds the desred change π n rsk neutral probabltes,.e., that π w + w R w π / π w+½ π D 2 π. Ths, n turn, can be verfed through substtuton by expressng the dpont value of the dervatve atrx, D π w + 2 w, as Πw + ½ ΠwRw Π w + ½ Π wrw, on the assupton that Rw s effectvely constant whch s equvalent to gnorng thrd-order effects. Q.E.D. The atrx R w, whch easures the decson aker s local rsk tolerance, plays a pronent role n these results. 3 Part a establshes that R w s the lnear transforaton that aps sall changes n ncoe nto changes n the optal dstrbuton of wealth; thus, the decson aker prefers to redstrbute ncoe across states n proporton to rsk tolerance. Part b establshes that R w s also the lnear transforaton that aps sall changes n relatve arket prces nto changes n the optal dstrbuton of wealth. It s suggestve to rewrte the forula n part b as follows: Rw w π π + w 2 + y. 6 π The ter on the left can be nterpreted as the change n wealth expressed n densonless, rskadjusted unts,.e., unts of wealth dvded by rsk tolerance. By vrtue of the result n part a, the constant ter y on the rght has no effect on the decson aker s rsk neutral probabltes, as 3 The forulas n the proposton are stated n ters of the nonsngular, but unobservable, atrx R, rather than ts observable but sngular counterpart R = R Π R. However, the forulas are nvarant to the addton of constants to coluns of R, so R could be replaced by R +A, where A s any atrx wth non-ero constant coluns. 23

25 t corresponds to a ove along the wealth expanson path assung Rw Rw+ w. Hence, the ter π / π w+½ π, whch s the negatve of the relatve change n prces, s entrely responsble for producng the requred change n the decson aker s rsk neutral probabltes. Now, ntutvely, the decson aker should react to oveents n prces by shftng wealth away fro states where the relatve prces of contngent clas have ncreased. The rewrtten forula 6 shows that equlbru wth the arket s restored by akng a change n rskadjusted wealth that s exactly equal and opposte to the relatve change n prces. The constant ter y erely re-centers the transacton so that t s self-fnancng at the new prces. When preferences are separable, the precedng results can be splfed by replacng the rsk averson atrx Rw wth the observable rsk averson vector rw. Correspondngly, the rsk tolerance atrx R w s replaced by the rsk tolerance vector tw rw. Corollary 4.: For a decson aker wth separable preferences n a coplete arket for contngent clas: a the change n wealth nduced by a sall lup-su ncoe x, holdng prces fxed, s w t w x, π [ t w] E w hence the rsk tolerance vector tw s the drecton of the wealth expanson path at w; and b the self-fnanced change n wealth nduced by a sall change π n arket rsk neutral probabltes s π Eπ w + π t w w π t w + y, where π w + 2 π y. π w + π 2 Eπ w + π [ t w ] 24

26 7. AN APPLICATION TO PROJECT VALUATION Proposton 4 and ts corollary are potentally applcable to the proble of choosng aong alternatve rsky projects n the settng of a coplete arket for contngent clas. A project s charactered by a strea of state- and te-dependent cash flows. The decson aker e.g., a fr s assued to have a utlty functon whose arguents are state- and te-dependent aounts of consupton, and asset purchases can be used to transfer consupton across te and states. If the arket s coplete, the soluton to the project selecton proble does not depend on the decson aker s own rsk preferences. Rather, the optal project s the one whose cash flow strea has the hghest expected net present value, where the arket rsk-free nterest rate s used for purposes of dscountng and the arket rsk neutral probabltes are used for coputng expected values. Ross 976, 978; Rubnsten 976 The decson aker s rsk preferences play a role only n the fnancng proble.e., gven the optal project, what addtonal asset purchases should be ade for optal borrowng and rsk hedgng? The fnancng proble s solved by purchasng assets so as to restore the equlbru between the decson aker s rsk neutral probabltes and those of the arket, after the project has been added to the exstng portfolo. If the decson aker s n equlbru wth the arket pror to the project decson, the optal fnancng decson s to short the project cash flows by sellng a replcatng portfolo and then nvest the arbtrage proft accordng to the forula n part a of the proposton or ts corollary. If the decson aker s not yet n equlbru wth the arket, the optal fnancng decson can be approxately deterned by applyng the forula n part b after addng the project to current wealth. The sae odelng fraework apples equally well to nterteporal decson probles, where wealth vares across te as well as states and the eleents of the vector π w are ore 25

27 approprately called noraled state prces rather than rsk neutral probabltes. As an exaple of an nterteporal applcaton, consder the sple captal budgetng proble due to Trgeorgs and Mason 987 and dscussed by Nau and McCardle 99 and Sth and Nau 995. There are two dates 0 and and two states of the world good and bad returns on nvestent at date. The fr has three decson alternatves wth respect to the constructon of new plant: nvest, defer, and declne. The fr can also buy or sell two securtes, one rsky and one rsk-free. The net cash flow streas assocated wth the alternatves and the securtes are as follows: Date 0 Date, good state Date, bad state Invest Defer Declne Buy share of rsk-free asset Buy share of rsky asset Let a consupton strea be represented by a correspondng 3-vector w 0, w g, w b. The fr s utlty functon s assued to have the te-addtve exponental for Uw 0, w g, w b = expw 0 / expw g / expw b /220 Thus, t s as f the fr assgns equal probablty to the two states at date and has constant absolute rsk averson wth rsk tolerances of 200 and 220 at dates 0 and respectvely. Fro the asset prces and the utlty functon, assung constant pror wealth, the followng state prces can be derved for the arket and for the fr, the latter dependng on the alternatve chosen: Noraled state prces Date 0 Date, good state Date, bad state Market Fr w/invest Fr w/defer Fr w/declne

28 Notce that the fr s state prce dstrbuton the noraled gradent of Uw s farly close to that of the arket under the Defer and Declne alternatves absent any asset purchases, but t devates sharply under the Invest alternatve. The optal overall strategy for the fr s to choose the alternatve whose cash flow strea has the hghest value under the arket s state prces and then purchase assets so as to equale the fr s state prces wth those of the arket. The followng table copares the arket values of the three alternatves, as well as the exactly- and approxately-optal asset purchases under each alternatve, where the approxaton forula fro Corollary 4.b has been used. The Defer alternatve s the optal choce, and the approxaton forula yelds asset quanttes very close to the exact values. Market value Optal shares of rsk-free asset Optal shares of rsky asset Noraled Un-Noraled 4 Exact Approx. Exact Approx. Invest Defer Declne The approxaton s slghtly less good for the Invest alternatve than for the others, because the necessary adjustent to state prces s uch larger n that case, but nevertheless the approxaton s so close that the buyng rsk preu for the dfference n wealth dstrbutons, whch easures the pact of the error n onetary ters, s only neglgble n coparson to the project value. If the fr s already n equlbru wth the arket.e., n possesson of the optal asset poston for the Declne alternatve part a of Corollary 4. can be appled nstead to deterne the change n the asset poston that s needed when choosng the Defer opton. The 4 The un-noraled value whch s the expected net present value at arket probabltes and dscount rates s obtaned by scalng the state prces so that the date 0 state prce s. 27

29 soluton s to sell the Defer alternatve at ts arket value.e., sell a replcatng portfolo, then nvest the ncoe x = 3.05 n noraled unts accordng to the forula n 4.a,.e., redstrbute the ncoe across the three te-state contngences n proporton to local rsk tolerances 200, 220, and 220, respectvely. Under constant absolute rsk averson, the latter ethod yelds an exact soluton to the fnancng proble, whereas the forula n 4.b yelds only an approxaton albet a very good one. The pont of ths exercse s to show that, for a decson aker wth separable preferences n a coplete arket, rsk neutral probabltes and statewse rsk tolerances are suffcent statstcs for choosng aong sall or oderate rsks and deternng optal hedgng strateges. Under ore general preferences, the rsk neutral probabltes and ther atrx of dervatves would be suffcent. 8. CONCLUDING COMMENTS The concepts of probablstc belefs, rskless wealth postons, and consequences wth statendependent utlty have tradtonally played key roles n odels of choce under uncertanty. However, a growng body of lterature casts doubt on ther unqueness and observablty, f not ther exstence, and consequently t s of nterest to deterne whether econoc phenoena such as averson to rsk and uncertanty can be odeled wthout reference to the. Yaar s payoff-convexty defnton of rsk averson, whch does not refer to probabltes or rskless wealth, suggests that a sple and general easure of rsk averson ought to be avalable for a broad class of preferences under uncertanty. Ths paper has derved such a easure n ters of the atrx of dervatves of the decson aker s local rsk neutral probabltes. The easure apples to farly general preferences, ncludng expected-utlty preferences that need not be state-ndependent and sooth non-expected-utlty preferences that need not be probablstcally sophstcated. It has been shown that varous aspects of rsk averse behavor and fnancal 28

30 decson akng can be odeled entrely n ters of the decson aker s rsk neutral probabltes, consstent wth the central role that rsk neutral probabltes play elsewhere n odels of arkets under uncertanty e.g., asset prcng by arbtrage. If the decson aker has separable preferences wth stochastc pror wealth and/or state-dependent utlty, her rsk neutral dstrbuton s the approprate dstrbuton wth whch to copute the varance n Pratt s rsk preu forula, and her local decson-akng behavor s copletely charactered by the rsk neutral dstrbuton and a vector easure of rsk averson. Under ore general sooth nonexpected utlty preferences, the rsk averson easure s atrx-valued and provdes a bass for coparng rsk averson between ndvduals wthout observng or agreeng on true probabltes and wthout reference to the 45-degree certanty lne, whose locaton ay be unknown. A copanon paper Nau 2002 shows how averson to uncertanty as well as rsk ay be encoded n the dervatves of the decson aker s rsk neutral probabltes. REFERENCES. ARROW, K.J. 964 The Role of Securtes n the Optal Allocaton of Rsk-Bearng. Quarterly J. Econocs 3, ARROW, K.J. 965 The Theory of Rsk Averson, Lecture 2 n Aspects of the Theory of Rsk-Bearng Yrjo Jahnsson Lectures, Yrjo Jahnssonn Saato, Helsnk. 3. AUMANN, R. 97 Letter to L.J. Savage. Reprnted n Drèe, J. 987 Essays on Econoc Decson under Uncertanty. Cabrdge Unversty Press, London. 4. CHEW, S.H. 995 A Schur-Concave Characteraton of Rsk Averson for Non-expected Utlty Preferences, J. Econ. Theory 67, CHEW, S.H., KARNI, E. and Z. SAFRA 987 Rsk Averson n the Theory of Expected Utlty wth Rank Dependent Probabltes, J. Econ. Theory 42, COHEN, M. 995 Rsk Averson Concepts n Expected and Non-expected Utlty Theory. Geneva Papers on Rsk and Insurance Theory 20:,